finite nuclei and nuclear matter in relativistic hartree-fock approach long wenhui 1,2, nguyen van...
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Finite Nuclei and Nuclear Matter in Relativistic Hartree-Fock Approach
Long Wenhui 1,2, Nguyen Van Giai 2, Meng Jie 1
1School of Physics, Peking University, China2Institut de Physique Nucleaire, Universite Paris-Sud, France
Contents
Introduction and motivations Theoretical Framework Numerical Calculations Results and Discussions Summary
Introduction Relativistic Hartree-Fock (RHF)
• Without self-interactionsA. Bouyssy, J.-F. Mathiot, N. V. Giai, S. Marcos, Phys. Rev. C36-380(1987).
• With -meson self-interactions P. Bernardos, V. N. Fomenko, N. V. Giai et. al., Phys. Rev. C48-2665(1993).
• With zero-range self-interactionsS. Marcos, L. N. Savushkin, V. N. Fomenko et. al., arXiv: nucl-th/0307063.
Advantage of RHF approach• More fundamental theory• Nuclear structure: spin-orbit interaction
Motivations
RMF theory and RHF approach• Hartree Hartree-Fock• Contributions of -meson• Pairing force in RHB theory
Proposal:• The effective interactions in RHF approach
PK1: PHYSICAL REVIEW C 69, 034319 (2004)
• The contributions of -meson• Different nonlinear mechanism
Lagrangian and Hamiltonian Lagrangian Density
Hamiltonian Density
35
2 2 2 2
3 422 3
1
2
1 1 1 1 1 1
2 2 4 2 4 21 1 1 1 1
2 2 4 3 4
fi M g g g e
m
m m R R m
m F F
A
g g
L
3
2 2 2 2
3 422 3
1
2
1 1 1
2 2 21 1 1 1
2 2 3 4
i ii i
ii
fi g g g e A
m
m m R m
F A m g g
H
Hartree-Fock Approach 0
Hartree-Fock Trial State
Expectations (see -meson as representative)
Fierz transformation (n=2, 3, 4)
†0
† † † †0
iE t iE t
iE t iE t
x f e c g e d
x f e c g e d
x x
x x
†0 0c
0 0E H
, ,n
s b Tf Fierz Transformation
Radial Dirac Equation Dirac Equation
G and F separations
0
0
0
0
T S
T S
dG G E M F X
dr r
dF F E M G Y
dr r
2 2 2 2G FG F
W W W WG F G F
0
0
0
0
G FT S
F GT S
dX G E M X F
dr r
dY F E M Y G
dr r
PKA 938.5 586.707 11.576 13.537 3.184 0.0 -38.962 41.967
HF(e) 938.9 440.0 7.2302 11.2100 2.6290 1.0027 0.0 0.0
HFSI 939.0 412.0 7.0942 11.4320 2.6290 1.0027 -67.18 -14.61
ZRL1 939.0 497.8 8.3695 12.1420 2.6290 1.0027 -29.646 51.00
M mg g g f 410b 310c
6 8
2 3783.0, 770.0, 138.0, ,g g
m m m g bM g cm m
Tab. I Effective Interactions
HF(e): (, , and )HF
A. Bouyssy, J.-F. Mathiot, N. V. Giai, S. Marcos, Phys. Rev. C36-380(1987).
HFSI: (, , and )HF + self-interactions
P. Bernardos, V. N. Fomenko, N. V. Giai et. al., Phys. Rev. C48-2665(1993).
ZRL1: (, , and )HF + zero-range self-interactionsS. Marcos, L. N. Savushkin, V. N. Fomenko et. al., arXiv: nucl-th/0307063.
Observables Nuclear Matter: 0, K, EB, asym
Binding energies of the following nuclei:
16O, 40Ca, 48Ca, 56Ni, 68Ni, 90Zr, 116Sn, 132Sn, 182Pb, 194Pb, 208Pb
0 EB K asym M*
Empirical data 0.166±0.018 -16.±1. 240±50 32.±8.
PKA 0.150 -15.99 276.15 29.51 0.54
HF(e) 0.149 -16.40 465 28 0.56
HFSI 0.14 -15.75 250 35.0 0.61
ZRL1 0.155 -16.39 250 35.0 0.58
Tab. II Bulk Properties of Nuclear Matter
Tab. III Binding energies and charge radii
16O 40Ca 48Ca 56Ni 68Ni 90Zr 116Sn 132Sn 182Pb 194Pb 208Pb
127.6 342.1 416.0 484.0 590.4 783.8 988.7 1102.9 1411.7 1525.9 1636.4
PKA 127.7 342.2 416.2 473.6 586.3 784.6 984.4 1103.5 1412.2 1526.2 1635.9
HF(e) 89.8 272.8 340.8 666.0 1401.9
HFSI 118.9 333.2 405.6 772.2 1618.2
ZRL1 117.9 333.2 408.5 780.3 1632.8
2.730 3.478 3.479 4.270 4.625 5.442 5.504
PKA 2.732 3.444 3.550 3.847 3.930 4.304 4.650 4.750 5.392 5.461 5.502
HF(e) 2.73 3.47 3.47 4.26 5.50
HFSI 2.73 3.48 3.48 4.26 5.52
ZRL1 2.71 3.44 3.49 4.25 5.49
Fig.1 The binding Energies of Pb isotopes
184 188 192 196 200 204 208 212
-6
-4
-2
0
2
4
6
A
Eex
p. -
Eca
l. (M
eV)
PKA PK1 NL3
Fig. 2 Single Particle Energies of 132Sn
Fig. 3 Single Particle Energies of 208Pb
Fig.4 Charge density distributions
0 1 2 3 4 5 6 7 8 9 100.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 1 2 3 4 5 6 7 80.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 1 2 3 4 5 60.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0 1 2 3 4 5 60.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0 1 2 3 4 50.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Exp. PKA PK1
208Pb
90Zr48Ca
40Ca
16O
Summary Programs for RHF approach are constructed New effective interaction PKA with -, -, -mesons
and nonlinear high order terms is obtained Better descriptions for nuclear matter and finite
nuclei are obtained. Perspective
• Isotopic shifts
• hard equation of state
• Contributions of -meson
• Density-dependent RHF
Thank you!
Sigma Field -meson field
Hamiltonian for -meson
4 ,x g d yD x y y y
2 3 4
0
1,
2 tH g x y D x y y x d xd y
2' '
; ' '
1,
2g f f D x y f f
H x y y x† †
' 'c c c c
0 0
', '
', '
D
E
EE H
E
2' '
; ' '
1,
2g f f D x y f f
H x y y x † †' 'c c c c
Multipole Expansion of the propagator
Potential Energy and Self-Energy
2 2 200
22 2 22
, 1 1 '0; 2 2
2 200
2 22
1 12 2
14 ' ' ' ; ',
2ˆˆ ˆ1
' ; ',02 4
' ' ' ; , '
ˆˆ
4
Ds s
Eq q Lr r
L
S s
q q
E g r r drdr r V m r r r
j j Lj j LE g drdr G G F F V m r r G G F F
g r dr r V m r r
j jj LX g
2
'0;
22 22
1 1 '0; 2 2
' ; ',0
ˆˆ' ; ',
04
LrL
q q LrL
Ldr G G F F V m r r F r
j j Lj LY g dr G G F F V m r r G r
1 12 2
0
1/ 2
1; , ' ; , ' '
4
1; , '
'
L L LL
L L L
eD r r V r r
V r r I r K rrr
Y Y r-r'
r - r'
-meson Pseudo-vector coupling
Exchange potential
5NN
f
m
L
2
2 21 2 1 2 1 22 2
1 13
3T c
fV
m m
V V
q q q q qq
q q
2 2
1 2 1 2 2 2
11
3c f m
Vm m
q
q
23
1 2 1 2 3
4
12
m rc m f e
Vm m m r
r r